414 ieee transactions on ultrasonics, ferroelectrics,...

414 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 64, NO. 2, FEBRUARY 2017

Bilateral CMUT Cells and Arrays: EquivalentCircuits, Diffraction Constants, and

Substrate ImpedanceHayrettin Köymen, Senior Member, IEEE, Abdullah Atalar, Fellow, IEEE, and A. Sinan Tasdelen

Abstract— We introduce the large-signal and small-signalequivalent circuit models for a capacitive micromachined ultra-sonic transducer (CMUT) cell, which has radiating plates onboth sides. We present the diffraction coefficient of baffled andunbaffled CMUT cells. We show that the substrate can bemodeled as a very thick radiating plate on one side, which can bereadily incorporated in the introduced model. In the limiting case,the reactance of this backing impedance is entirely compliantfor substrate materials with a Poisson’s ratio less than 1/3.We assess the dependence of the radiation performance of thefront plate on the thickness of the back plate by simulating anarray of bilateral CMUT cells. We find that the small-signallinear model is sufficiently accurate for large-signal excitation,for the purpose of the determining the fundamental component.To determine harmonic distortion, the large-signal model mustbe used with harmonic balance analysis. Rayleigh–Bloch wavesare excited at the front and back surfaces similar to conventionalCMUT arrays.

Index Terms— Backing impedance, bilateral, capacitive micro-machined ultrasonic transducer (CMUT), diffraction constant,equivalent circuit model, hydrophone, transducer, two sided.

I. INTRODUCTION

THE capacitive micromachined ultrasonic transducer(CMUT) technology is more than 20 years old [1]. There

had been impressive progress in the theory of capacitivetransducers, in the modeling approaches, both numerical andanalytical, and particularly in the production technologiesduring these two decades. The CMUT technology is compara-tively assessed with respect to other conventional technologiesduring this period. Despite the fact that it is a mature tech-nology, there still remain quite few issues to be addressed,either because they are specific to CMUTs or because theenabling nature and superior performance potential of CMUTtechnology only recently brought these issues forward, whichwere unnoticed with conventional transduction approaches.Some of such issues, namely, the bilateral CMUT cell, an ana-lytical equivalent circuit for this cell, an analytical model for

Manuscript received April 7, 2016; accepted November 10, 2016. Date ofpublication November 15, 2016; date of current version February 1, 2017. Thiswork was supported by the Scientific and Technological Research Council ofTurkey under Grant 114E588. The work of A. Atalar was supported by theTurkish Academy of Sciences.

H. Köymen and A. Atalar are with the Department of Electrical andElectronics Engineering, Bilkent University, 06800 Ankara, Turkey (e-mail:[email protected]).

A. S. Tasdelen is with the Bilkent University Acoustics and UnderwaterTechnologies Research Center, Bilkent University, 06800 Ankara, Turkey.

Digital Object Identifier 10.1109/TUFFC.2016.2628882

Fig. 1. Cross-sectional view of a bilateral circular CMUT cell. The radiationplates are used as the electrodes as shown, assuming conducting plate materiallike doped silicon.

the acoustic termination of the gap in conventional CMUTcell, and the diffraction constant of conventional and bilateralCMUT, are presented in this paper.

The applications of CMUT, hence the equivalent circuitmodels, are confined to cells or arrays radiating into the half-space, where the substrate is assumed to be rigid. The extentto which the substrate can be considered rigid depends onthe various parameters like the stiffness and the thickness ofthe substrate material, the backing material that follows thesubstrate, as well as the relative size of the CMUT cell or thearray. Since the substrate is of finite thickness and is made ofmaterials with finite stiffness and density, a CMUT cell canbe considered as a vacuum gap with two vibrating plates onboth sides, as depicted in Fig. 1. Such a bilateral CMUT cantransmit into and receive from both half-spaces. In this paper,we present three-port equivalent circuit model for bilateralCMUT cells, which is an extension of the model given in [2].

We used this accurate model of the bilateral CMUT cell toderive the acoustic impedance at the back side of the gap of aconventional CMUT cell. The acoustic termination at the backside of the gap in a conventional CMUT cell has significanteffects, particularly on the received signal. This effect oftenmanifests itself as a ringing [3], [4]. The effects of substrateand other backing material when the size of the cell or anarray of cells is large compared with the wavelength had beenstudied extensively [5]. The presented model is instrumentalin analysis and quantification of the effect of the substratematerial and its thickness on the backing impedance in CMUTcells. In Section III, we study the interaction of the bottom

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KÖYMEN et al.: BILATERAL CMUT CELLS AND ARRAYS 415

surface of the gap with the substrate and propose an analyticallumped model for the backing impedance presented by thesubstrate to the conventional CMUT cell when the size of thecell is small compared with the wavelength.

A CMUT cell with two moving plates, similar to thisstructure but radiating to one hemisphere, is proposed recently[6]–[8]. This new structure is referred to as “multiple movingmembrane” (M3 CMUT) and claimed to have larger swingin the upper plate at a lower dc voltage bias compared witha similar sized conventional CMUT, thus promising betterperformance for airborne applications. This new structureis fabricated employing a MEMSCAP sacrificial technique,PolyMUMPs [9]. The production support of this well-established and popular multiuser foundry service is a veryimportant advantage for such CMUT designs using two vibrat-ing plates. It is shown in this paper that the presented modelprovides a means for very accurate analysis and design ofM3 CMUT.

It is shown in this paper that the bilateral CMUT celloffers advantageous applications such as omnidirectional freefield microphones, hydrophones, and sources. A very highsensitivity over a very wide bandwidth is obtained in immer-sion for a given size, when operated at resonance or offresonance, compared with other transduction mechanisms.The bilateral CMUT is comprehensively analyzed using thepresented model.

The transmission and reception characteristics of any trans-ducer require the knowledge of relevant diffraction constants.Since CMUT cells are conventionally considered in rigidbaffle, the diffraction constant is always equal to two [10]for normally incident waves. The diffraction constants forbaffled and unbaffled conventional and bilateral CMUT cellsare presented in Section IV, both for transmission and forreception.

Bilateral CMUT cell and arrays are simulated and the effectsof different thickness of radiating plates are evaluated inSection V.

II. BILATERAL CMUT CELL MODEL

The cross-sectional view of a bilateral circular CMUT cellis shown in Fig. 1, where a is the diameter of the CMUT,tg is the gap height, ti is the insulating layer thickness, and tmand tm are the front and back plate thicknesses, respectively.The immersion medium on front and back surfaces may bedifferent. Acoustic waves are generated in the front and backmedia by applying an electrical signal between the conduct-ing plates acting as electrodes of the CMUT. The referencedirections of both displacements, xP and x P , are toward thegap. It is assumed that the plates are perfectly clamped, haveuniform thickness, and operated in their mechanically linearregion, both material wise and geometrically, as in [2].

A. Transduction Section Model and Biasing

We assume that the front and back surfaces have uncol-lapsed deflection profiles given by

x(r, t) = x P(t)

(1 − r2

a2

)2

; x(r, t) = x P(t)

(1 − r2

a2

)2

(1)

where xP(t) and x P(t) are the center displacements of thefront and back plates, and r is the radial distance from thecenter as shown in Fig. 1. Letting ε0 show the permittivity ofthe free space, the capacitance C(t) between the plates can bewritten as an integral

C(t) =∫ a

0

ε02πr

tge − x(r, t) − x(r, t)dr

=∫ a

0

ε02πr

tge − (x P(t) + x P(t))(1 − r2

a2

)2 dr (2)

where tge = tg + ti/εr is the effective gap height and εr is therelative permittivity of the insulator. Comparing this equationwith [2, eq. (2)], we see that every analytical result in [2]regarding the equivalent circuit model of transduction sectionis valid for the bilateral cell, if we set the total normalizedcenter displacement as

u(t) = xP(t) + x P(t)

tge. (3)

The deflected clamp capacitance becomes

C(t) = C0g(u(t)) (4)

where C0 = ε0πa2/tge is the undeflected clamp capacitancebetween two plates and g(u) is the transduction function asgiven in [2, eq. (4)].

Energy and transduction force expressions are again thesame as in [2]. We define the static forces due to the ambientpressure acting on the two radiating surfaces of the CMUT cellas FPb and FPb . These two forces are equal, if the ambientpressure is the same for two faces. The compliances of theplates to be used with the peak model are CPm and CPm ,given as

CPm = 9(1 − σ 2)a2

16πY0t3m

(5)

and

CPm = 9(1 − σ 2)a2

16π Y0 t3m

(6)

where (Y0, σ ) and (Y0, σ ) are the Young’s moduli andPoisson’s ratio of the front and back plate materials,respectively [2]. The respective peak static displacements,X P and X P , are obtained as

X P = CPm

(fP

(X P

tge+ X P

tge

)+ FPb

)(7)

X P = CPm

(fP

(X P

tge+ X P

tge

)+ FPb

)(8)

where fP is the electrostatic force function between theelectrodes as defined in [2, eq. (7) and Table I], suitablefor spatial peak quantities. The total normalized displacementbecomes

X P

tge+ X P

tge= CPm + CPm

tgefP

(X P

tge+ X P

tge

)+ FP B

FPG(9)


Fig. 2. Large-signal nonlinear equivalent circuit model for the bilateralCMUT cell. The velocities vR and vR and force fR in the mechanical sectionare spatial rms variables. Z RR and Z RR are the radiation impedances suitablefor the rms variables.

where

FP B = CPm FPb + CPm FPb

CPm + CPm(10)

is the effective static force due to ambient pressure and

FPG = tge

CPm + CPm(11)

is the force required to have the two center displacements sumup to tge.

The collapse voltage in vacuum Vr is given by

Vr =√

4t2ge

3C0

(1

CPm + CPm

). (12)

The reference voltage, Vr , is lower compared with a one-sidedCMUT cell as expected. CMUT biasing chart (CBC)1 equationbecomes

VDC

Vr=

√√√√√3(

X P +X Ptge

− FP BFPG

)

2g′(

X P +X Ptge

) forX P + X P

tge≥ FP B

FPG. (13)

The variation of Vdc with respect to total normalizeddisplacement is the same as for one-sided cell. The resultsobtained in [2] are directly applicable.

B. Large- and Small-Signal Equivalent Circuit

The large-signal rms equivalent circuit model [2] with thechosen directions for motion is shown in Fig. 2. The circuitparameters are exactly the same as given in [2, Table I].2

With the same immersion media on both sides, the radiationimpedances Z R R and Z R R are equal to each other. Thecompliance of rms equivalent circuit is 1/5 of the complianceof the peak circuit, CRm = CPm/5, and CRm = CPm/5 [2].Similarly, the inductances in the mechanical section are themasses of the front and back plates.

The small-signal equivalent circuit, shown in Fig. 3, is sim-ilar to the circuit given in [2, Fig. 6], except that the twobranches for two radiating faces emerge after −CRS, the spring

1The static equilibrium for parallel plate capacitor was studied, and theequilibrium equation was plotted and discussed as equilibrium curve in [11].The equilibrium for clamped circular plate of CMUT was derived and plottedin [2], which was later referred to as CBC.

2 pin in calculating f I in [2, Table I] must be taken as 2pin instead of pinfor waves perpendicularly incident on the CMUT surface since the modelassumes that the CMUT cell is in rigid baffle [10].

Fig. 3. Small-signal linear model of the bilateral CMUT cell. n R is the turnsratio of the electromechanical transformer suitable for the peak variables.

softening capacitor. The variables nR , CRS, and C0d arecalculated as

nR = 2FR

VDC(14)

CRS = 2t2ge

5C00V 2DCg′′

(X P/tge + X P/tge

)

=(

CRm + CRm

) [2

3

V 2DC

V 2r

g′′(

X P

tge+ X P

tge

)]−1

(15)

C0d = C0g

(X P

tge+ X P

tge

)(16)

respectively, where the electrostatic force is given as

FR = √5

C0V 2DC

2tgeg

′(

X P

tge+ X P

tge

). (17)

C. Validation of the Model

The finite-element analysis (FEA) results of an M3 CMUTwith 35-μm radius are reported in [8]. The plate thicknessesand gap height, which are the layer dimensions in Poly-MUMPs process, are given in [8, Table 1]. The materialproperties of polysilicon are also given right after Table 1 asY0 = 158 ± 10 GPa, σ = 0.22 ± 0.01, and ρ = 2328 kg/m3.The analytical model presented in this paper is validated withthe FEA results reported in [8]. The relevant data for modelcalculations are determined from [8] as tge = 750 ± 80 nm,tm = 1.5 ± 0.1 μm, and tm = 2 ± 0.15 μm.

Using the median material and geometry data, the compli-ances are found as CRm = 7.83 × 10−5 m/Nt and CRm =3.30 × 10−5 m/Nt, and Vr as 172.2 V. It is pointed out thatthe release holes were ignored in FEA and the plates areassumed initially flat, i.e., Fb/Fg = 0. With 135 V bias,Vdc/Vr = 0.784, and CBC, (13) yields X P + X P = 0.1707tgefor this bias level. The static transduction force, FR , is found as5.12 ×10−4 Nt under same conditions, using (17). The centerdeflections of the front and back plates are determined as90 and 38 nm, respectively, using (7) and (8), which matchesto the FEA results reported in [8, Fig. 3]. The calculations arerepeated for bias voltages changing between 10 and 150 V,and the FEA results reported in [8, Fig. 4] are obtained fromthe model.

The masses of the plates (inductances in mechanical section)are found as 1.34 × 10−11 and 1.79 × 10−11 kg for L Rm andL Rm , respectively. The mechanical resonance frequencies of


the front and back plates are found in vacuum as 4.907 and6.543 MHz, respectively, using the respective plate compli-ances and masses. When combined in series with −CRS,which is −4.58 × 10−4 m/Nt at 135 V bias, the combinedmechanical section on the right of electromechanical trans-former in Fig. 3 has a resonance at 4.397 MHz.

The two mechanical resonators are coupled through thetransduction force generated in the gap, since the displacementin each branch adds up to modify the force. This phenomenondominates the transient response, unlike the static analysis,when there is no mechanical loss in the system. The Q factorsof mechanical branches are more than 200 at 4.397 MHzif the only loss is radiation resistance. Loss mechanismslike radiation resistance at each acoustic port or frictionalresistance within the CMUT cell decrease the displacementamplitude in each plate and mitigate the effects arising fromcoupling at steady state. The time required to reach steadystate is related to the time constants of the system and can bevery large for low radiation resistance levels.

Transient response of lossless airborne CMUT cannotbe usefully evaluated using FEA because of long tran-sient responses and convergence problems while it is astraightforward circuit simulation in lumped element models.Large-signal equivalent circuit is implemented in ADS3 circuitsimulator, and the transient response of the same M3 CMUTcell is simulated at 4 MHz. The response reaches to steadystate in about 100 μs, whereas only 1.5 μs of transientresponse is given in [8]. It is observed that the nonlinear effectsbecome dominant at drive voltage levels exceeding 1 V peak,where 40-nm swing is already generated in the upper plate,and collapse occurs at about 7.5 V. The exact conditions andprecautions, like introduced loss for convergence, used in FEAanalysis are not described in [8]. When a significant amountof extra loss in addition to radiation resistance is included inthe equivalent circuit, transient response with amplitude levelssimilar to the one depicted in [8, Fig. 5] is observed. The firstfew cycles in the initial 1.5 μs of the transient response thatare depicted in [8, Fig. 5] can only be qualitatively assessedto be in agreement.

The plates produced in PolyMUMPs process have releaseholes. The release holes are ignored in the FEA in [8] and inour analysis. The release holes have two effects on the plate:1) the compliances of the plates increase, which is effectivein static and dynamic performance and 2) they cause a verysignificant loss due to air friction, compared with radiationresistance during dynamic operation. The experimental mea-surements have lower collapse voltage and lower resonancefrequencies compared with FEA prediction due to the effectgiven above in 1) and a wider bandwidth and reasonably shorttransient response in dynamic analysis due to extra loss effectdescribed in 2).

D. Coupling Coefficient

The coupling coefficient in bilateral CMUT cell can read-ily be obtained using the approach given in [12]. When

3Advanced Design System, Agilent Technologies, Inc., Santa Clara, CA,USA.

the coupling coefficient definition of the square root of theof ratio of converted energy to the stored energy givenin [12, eq. (4)] is adopted, dynamic coupling coefficientbecomes

k = 1√1 + C0d

n2R

CRS−(CRm+CRm )

CRS(CRm+CRm )

(18)

in the absence of any parasitic capacitance, using the small-signal equivalent circuit. If there is parasitic capacitance to betaken into account, C0d must be augmented accordingly as in[12, eq. (8)]. The coupling coefficient increases to unity at thecollapse threshold.

III. SUBSTRATE AS BACKING IMPEDANCE

The plate on one face is designed for radiation in a single-sided CMUT cell, whereas the other is the substrate, whichis a very thick layer of a hard material. The impedancepresented to the equivalent circuit by this face can bedetermined by increasing the thickness of the correspond-ing plate. If the thickness of the plate at the back (bot-tom) face is gradually increased, the compliance, CRm , andthe displacement, xR , in that branch decrease. The compli-ance, CRm , becomes very small and dominates the branchimpedance.

If xR becomes insignificant compared with xR , we canconsider that a rigid backing terminates the gap and removethe branch from the circuit. However, xR can never be zerowith real materials. The limiting case is an infinitely thicksubstrate.

Assume we use a block of some solid material and aCMUT is implemented on this block, which is the commonassumption in CMUT models, analytic or numerical. Theradiating plate is on one side and a substrate on the other.

A CMUT cell radiating into air or water has significantlysmaller relative aperture on the substrate side. For example,consider an airborne CMUT cell with a reasonably large aper-ture, ka = 3, build on borosilicate glass [13]. The velocitiesof sound in air and in borosilicate glass (longitudinal) are340 and 5590 m/s, respectively. As far as the aperture onborosilicate glass is considered, ka becomes 0.18. This is like apoint source radiating spherical waves into a solid half-space.

The acoustic impedance of spherical waves in isotropicsolids is studied in [14]. Blake [14] derives the impedance as

Z Rb = Sρbcb{R1b(ka) + j X1b(ka)} (19)

where ρb and cb are the density and the velocity of sound inthe solid, S is the area of the radiating surface, and a is theradius. The normalized radiation resistance is given by

R1b(ka) = (ka)2

1 + (ka)2 (20)

similar to R1(ka) of fluid medium. The reactance is signifi-cantly different and is derived as

X1b(ka) = −1 + (1 − K )(ka)2

K (ka)[1 + (ka)2] (21)


Fig. 4. Normalized radiation resistance and reactance of spherical waves insolids. The reactance is plotted for different values of the Poisson’s ratio.

where

K = 1

2

1 − σ

1 − 2σ(22)

and σ is the Poisson’s ratio in the solid. The radiationresistance and reactance are plotted with respect to ka in Fig. 4,for different values of Poisson’s ratio.

When σ is in vicinity of 0.5, which means that the mediumis liquid, radiation reactance variation with respect to ka issimilar to that of water or air. Poisson’s ratio is close to0.5 also in soft polymers. Reactance is compliant only for verysmall values of ka and becomes similar to fluid for larger ka(σ = 0.48 depicted in Fig. 4). Reactance assumes negativevalues for small ka values and positive for larger, when1/3 < σ < 0.5. Note that the radiation impedance has aseries resonance at a particular value of ka in this region.

The reactance is always compliant when σ < 1/3. Poisson’sratio for borosilicate glass is 0.2, and therefore yields acompliant reactance for all values of ka.

The reactance, X1b (ka), can be decomposed as

X1b(ka) = ka

1+(ka)2 − 1

K

1

ka= X1(ka) − 1

K ac

1

ω(23)

where

X1(ka) = ka

1 + (ka)2 (24)

which is the same as that of fluid. The backing impedance cannow be written as

Z Rb = SρcR1(ka) + j SρcX1(ka) + 1

jω

Sρc

K ac

(25)

where Z Rb is the series combination of a component similarto the radiation impedance similar to that of a fluid (the samemorphology) and a compliance, where

CRb = K a

Sρc2 = (1 + σb)

2πaY0b(26)

is the series compliance, where Y0b and σb are the Young’smodulus and Poisson’s ratio of the substrate material, respec-tively. The equivalent circuit including the backing impedanceis shown in Fig. 5.

Fig. 5. Equivalent circuit model with infinite solid backing.

For an airborne CMUT cell, the reactance of Z Rb is verylarge compared with the impedance of the front (air) segment.Hence, the rigid substrate assumption is justified.

The compliance of a clamped plate in CMUT [2] is verylarge when compared with CRb in the backing impedance. Theratio of two compliances can be written as

CRm

CRb= 9(1 − σ 2)

40(1 + σb)

Y0b

Y0

a3

t3m

fora

tm> 8 (27)

where Y0 and σ are the Young’s modulus and Poisson’s ratioof the plate, respectively. The plate compliance can be morethan 100 times larger even for very thick plates with a/tm = 8,for plates and substrate made of the same material. The rigidbacking assumption is well justified for an airborne CMUTwith thick substrate.

IV. SYMMETRIC BILATERAL CMUT CELL

A. Unbaffled CMUT

When the radiation plate on either side is the same,the bilateral CMUT becomes symmetrical. The symmetricbilateral CMUT radiates in exactly the same way at eitherside, except the direction of radiation is opposite. Therefore,a full symmetry is established in the acoustic fields of twohalf-spaces on either side of the CMUT’s midplane.

As part of this symmetry, the normal component of particlevelocity must vanish on the midplane, which is the rigidboundary condition. Even if the CMUTs are not in a baffle,midplane behaves as a rigid baffle. The practical designconstraints ensure that the sum of the plate thickness and theinsulation layer thickness, and the gap height is very smallcompared with the wavelength in the immersion medium atthe operation frequency. Acoustically both radiating faces ofthe bilateral cell can be considered to be on the midplanebecause of this dimensional property

2tm + 2ti + tg � λ. (28)

Hence, both faces can be considered in rigid baffle, even whenthere is no baffle at all. The same arguments are also valid forsymmetric bilateral CMUT arrays if all cells are symmetricallydriven.

The model presented in this paper is also accurate for suchsymmetric bilateral CMUT cells and arrays without any rigidbaffle.

The pressure directivity pattern D(θ) of a CMUT cell inrigid baffle is given as [15], [16]

D(θ, ka) = 48J3(kasin(θ))

(kasin(θ))3 (29)


Fig. 6. Directivity pattern of symmetric bilateral CMUT for ka = 1.2, 2.4,3.6, and 4.8. Radial scale is logarithmic, in decibels.

where θ is the direction of propagation relative to the normal tothe plane of the cell. The same directivity pattern can be usedfor symmetric bilateral cells in two half-spaces. The directivitypattern is shown in Fig. 6 for various values of ka. Thedirectivity pattern is quite omnidirectional even for ka as largeas 2.4. The on-axis radiation is only 3 dB higher comparedwith the radiation in the midplane. Waterborne CMUTs canbe designed to be resonant at very low ka values, unlike, forexample, spherical PZT transducers.

B. Diffraction Constant of CMUT in Rigid Baffle

Transducers with uniform surface velocity are extensivelystudied [10], [17]–[21]. It is known that the diffraction con-stant, radiation resistance, and the directivity factor of suchtransducers are interrelated [10]. In [18], it is shown that thesame relation is also satisfied by transducers with nonuniformbut fixed surface velocity distribution. However, the lumpedsurface velocity, referred to as “reference” velocity, must becarefully defined in this case. Following the approach in [10]and [18], this relation for circular CMUT cell in rigid baffleis derived as:

D2a(ka) = 4π RR A(ka)D f

ρck2(πa2)2 = 4(1.8)R1(ka)D f

(ka)2 (30)

where Da is the diffraction constant, RR A and R1 are theradiation resistance defined for average velocity given in [16],[22], and [23] as

RR A(ka) = 1.8(πa2ρc)R1(ka) (31)

and

R1(ka) = 1 − 200

(ka)9 F1(2ka) (32)

and D f is the directivity factor of the CMUT cell, respectively.D f is calculated using the pressure directivity pattern, D(θ),given in (29). The directivity factor can be written as

D f = D f (θ, ka)|θ=0 = D2(θ, ka)2π4π

∫ π/20 D2(γ, ka)sinγ dγ

∣∣∣∣θ=0

.

(33)

Fig. 7. Directional plane wave diffraction constant for a CMUT cell inrigid baffle. (a) Diffraction constant versus ka for incidence angle θ = 0°(normal), 27°, 45°, and 90°. (b) Diffraction constant versus incidence anglefor ka = 0.25, 1.6, 3, 4, and 6.28.

Da(ka) turns out to be equal to two for all values of ka,as expected. The directional plane wave diffraction constantcan be calculated following the definition and the approachgiven in [19] as:

D2a(θ, ka) = 4(1.8)R1(ka)D f (θ, ka)

(ka)2 . (34)

The diffraction constant is depicted in Fig. 7 for differentvalues of ka and incidence angle. The diffraction coefficientis close to two for small ka for all incidence angles. Theminimum value of diffraction constant remains near two formoderate ka values in CMUT cell. For example, Da > 1.75for ka = π/2.

This is the transmission diffraction constant and is numer-ically equal to the receiving diffraction constant for cells inrigid baffle. The force applied on the CMUT surface due toincident pressure pi is given as

f I = Da(θ, ka)πa2 pi . (35)

The transmitting diffraction coefficient is also the same forsymmetric bilateral cells in two half-spaces.


C. Receiving Unbaffled Symmetrical Bilateral CMUT

When used as a hydrophone or a microphone, the lack ofrigid baffle must be taken into account. The pressure directivitypattern and the radiation resistance of the unbaffled symmetricbilateral CMUT are required in order to calculate the relevantreceiving diffraction coefficient. These two parameters arenot readily available analytically, but can be numericallyevaluated. However, an approximate measure can be definedby the help of prior work on thin disk hydrophones. Thediffraction constant of thin rigid disk had been studied asearly as 1932 for small ka [24]. Spence [25] extended thiswork for analytic calculation of diffraction constant for allka values. Woollett [18] calculated the diffraction constant offlexural disk type pressure gradient transducers including theeffect of circular inertial frame that holds the disk. Only thescattered field contributes to transduction in such transducers,and the diffraction constant is about 0.5 ka for small ka ifthe frame diameter is close to the disk diameter. Diffractioncoefficient increases for the same ka as the frame diameterbecomes larger.

On the other hand, it is known that Da of an unbaffled pistontransducer mounted to the end of a long rigid rod is unity forka � 1 [20]. As ka increases, Da also monotonically increasesto its upper limit of two for very large ka. Da is a measureof how much of incident pressure a receiving transducer canmake use of. For example, the diffraction constant of sphericaltransducers is also unity for very small ka, but falls off rapidlyas ka increases [20].

The velocity profile on the CMUT’s radiation plate isclamped disk profile and results in higher ka values forequivalent radiation resistance and pressure directivity levels,compared with uniform and simply supported profiles, exam-ined in the literature for diffraction coefficient. An accurateapproximate receiving diffraction constant, Dar (θ, ka), can beconjectured as

Dar (θ, ka) ≈{

Da(θ, ka), for ka > 3

1, for ka < 1.6(36)

A bilinear interpolation can be employed between 1 andDa(θ, ka), the transmitting diffraction pattern, for 1.6<ka <3and for different values of θ . It is important to note thatdiffraction constant can be taken as unity for ka < 1.6 inall directions, for unbaffled symmetric bilateral CMUT cell.The forces applied on the surfaces become

fI ≈ f I ≈ πa2 pi (37)

in this useful ka range.

V. ARRAY OF BILATERAL CMUT CELLS

A. Small-Signal Equivalent Circuit for anArray of Bilateral CMUT Cells

We consider N bilateral cells connected to form an arrayof m elements. Array elements are excited with voltagesources, Vin1, Vin2, . . . , V inN, each with a source resistanceof RT . The series mechanical loss resistances, RA and RA,are included into each mechanical branch to model the fric-tional or other loss mechanisms, which limit the amplitude

Fig. 8. Small-signal linear model of an array of bilateral CMUT cells.

of spurious Rayleigh–Bloch (RB) wave resonance vibra-tions [26]. As shown in Fig. 8, the element currents areiin1, iin2, . . . , iinm, while the individual cell currents arei1, i2, . . . , iN . The front and back plate peak velocities areshown with vR1, vR2, . . . , vRN and vR1,vR2, . . . , vRN , respec-tively. The radiation impedance matrices for front Z and backsurfaces Z are uncoupled, since the front and back immersionmedia are not acoustically coupled in an array of bilateral cells.With a sinusoidal excitation of the elements, 2N unknownplate velocities at a given frequency can be solved by theinversion of a 2N ×2N matrix, similar to what is done in [26].

B. Effect of Back Plate Thickness on Reception Performance

A single-element array of 7 × 7 cells, each with a radiusa = 22 μm and a front plate thickness tm of 1.5 μm (a/tm =14.7), is simulated in water for three different back platethicknesses, 2, 3, and 4 μm (a/tm = 11, 7.3, 5.5). The small-signal equivalent circuit is used for the simulation of receivingcells, since the small-signal conditions are satisfied both on theelectrical and the mechanical variables during reception. Theplate constants are Y0 = 149 GPa, ρ = 2370, and σ = 0.17.The effective gap height is 95 nm and the cells are biased at80% of their respective collapse voltages, where the standardatmospheric pressure is assumed for ambient pressure.

The short-circuit received current sensitivity in A/Pa isplotted in Fig. 9 for three back plate thicknesses. The seriesloss resistance in both branches is assumed to be RA =0.03πa2Zw [26], where Zw is the specific acoustic impedanceof water (1.5×106 Nt/m3/s). This amount of series mechanicalloss causes a decrease of 1 dB in the output current in allthree devices. Even with such a loss, the RB wave resonanceeffects are still clearly observable at frequencies below 5 MHz.When the arrays are operated in pulsed mode, the effect ofthese resonances is extended low amplitude ringing in timedomain [26], the duration of which is determined by thequality of the resonance.

The reception sensitivity slightly improves (less than 2 dB)for thicker back plate due to larger bias voltage, since thecells with thicker back plate have larger collapse voltages.If the same bias voltage of 13 V, 0.8 VC of the CMUT


Fig. 9. Received short-circuit current response of a bilateral waterborneCMUT element of 7×7 cells with differing back plate thicknesses, when 1-Paamplitude pressure wave is incident on the front surface. The cell dimensionsare a = 22 μm, tm = 1.5 μm and tge = 95 nm. The spacing between thecells, s, is 3 μm, ambient pressure, P0, is SAP, and Vdc = 0.8 VC . Thenormalized series loss resistance RA is taken as 0.03.

Fig. 10. Radiated power from the center cell of 7×7 element when the bias is0.8 VC and drive voltage is 1.8 V peak. The cell dimensions are a = 22 μm,tm = 1.5 μm, tm = 2 μm, and tge = 95 nm. The spacing between thecells s is 3 μm, and ambient pressure P0 is SAP. The normalized series lossresistance RA is taken as 0.03.

with 2-μm back plate, is used for all cells, the array withthinner back plate is most sensitive, 1.5 dB higher than theresponse of the thickest one. When the back plate is replacedby rigid substrate, reception performance is similar to that of4-μm-thick back plate in the frequency band of interest.

C. Effect of Back Plate Thickness on Radiation Performance

The ac drive voltage and the mechanical displacement ofthe plate usually exceed the limits of small-signal conditionsin transmitting CMUTs. The large-signal model is requiredfor transmission simulations, especially if the amount ofharmonic distortion is to be determined. The transmit responseis also affected by RB wave resonances, which causes furtherambiguity on the response when combined by the nonlinearnature of the CMUT.

In Fig. 10, the power radiated by the center cell of the 7×7array element with 2-μm-thick back plate is depicted whenthe bias voltage is 0.8 VC and ac drive voltage is a 1.8 V

Fig. 11. Time waveform of the force and the velocity at front and backplates of the center cell at 1.7 MHz.

Fig. 12. Radiated power to the front and back half-space by a bilateralwaterborne CMUT element of 7×7 cells with differing back plate thicknesses.

amplitude sine wave. The CMUT cell’s response is highlynonlinear under this drive condition. The fundamental compo-nent power for front and back plates and the total harmonicdistortion (THD) obtained by harmonic balance analysis usingADS circuit simulation software are plotted together with theresponse using small-signal model. The power in the funda-mental component follows the linear response very closely atall frequencies both for front and back plates, although thereis very significant amount of THD, particularly at the backplate. At 3.6 MHz, the THD is more than the fundamentalpower. THD is dominated by the second harmonic power, anda highly resonant behavior in THD is clearly observed at thefrequency range, which is half of that of RB wave resonancesoccur in linear operation.

The velocity of the front and back plates of the center celland force on the plates at 1.7 MHz are shown in Fig. 11.The nonlinear distortion in all signals is significant.

Radiated power to the front and back half-space is plottedin Fig. 12 using the small-signal model. The cells are biased


again at 80% of their respective collapse voltages and drivenby a sinusoidal signal of 500-mV amplitude from a 50-source (available power is −2 dBm). The RB wave resonancesare visible in the front radiated power at frequencies below5 MHz albeit at a much lower amplitude. Similarly, RB waveresonances exist at the back surface (e.g., at 6.9 MHz fortm = 2 μm). Note that these resonances weakly couple to theopposing surface. Since the total radiated power is contributedby all CMUT cells behaving differently, the strong RB waveresonances of the single cell seen in Fig. 10 tend to bewashed out. There is about 25-dB difference at 5 MHz betweenthe front and back radiated power in the CMUT with thethickest back plate. When the back plate is replaced by a rigidsubstrate, the front radiation is similar to that of CMUTs withthe thickest back plate, but radiation to the back hemispherevanishes.

VI. CONCLUSION

In this paper, we present the bilateral CMUT cell. A three-port equivalent circuit model is derived, and the effect ofthe back plate thickness is studied using this model. It isobserved that the emission from the back plate is 25 dBlower in the passband for waterborne operation, when theback plate is 2.7 times thicker compared with the radiatingplate and when the CMUT is biased at 80% of the collapsevoltage.

Very high receiving sensitivity of CMUTs makes the unbaf-fled symmetric bilateral CMUT, which is also presented in thispaper, a potential contender for omnidirectional hydrophoneapplications. Bilateral CMUT offers high sensitivity with asmall dimension compared with wavelength. CMUT cells canbe designed for wide-bandwidth operation at resonance at verylow ka values. This option makes CMUT hydrophones suitablefor high-sensitivity hydrophone operation both at resonanceand off resonance. Using symmetric bilateral CMUT in col-lapsed mode will provide a tradeoff between the sensitivityand ambient static pressure tolerance.

It is shown in this paper that the energy lost to the substratecan be ignored in airborne CMUTs with thick substrate,since the backing is essentially rigid. However, there maybe multiple reflections of this very low energy due finitedimensions of the substrate, which may yield ringing. Thislow amplitude ringing interferes with the received signalin pulse-echo applications and limits the dynamic range.Using some absorbing material in contact with the sub-strate is instrumental to avoid the multiple reflections in thesubstrate.

The linear model is sufficient to characterize thefundamental component of the response, even at high exci-tation levels. Nonlinear model with harmonic balance sim-ulation is needed to determine the level of the harmonicdistortion components. RB wave resonances cause significanteffects in the individual cell responses even in the presenceof mechanical loss in the plates. However, their effect isreduced and averaged out when the outputs of many cells aresummed as in the case of parallel connected cells forming anelement.

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Hayrettin Köymen (S’74–M’76–SM’91) receivedthe B.Sc. and M.Sc. degrees in electrical engineeringfrom Middle East Technical University (METU),Ankara, Turkey, in 1973 and 1976, respectively,and the Ph.D. degree in electrical engineeringfrom Birmingham University, Birmingham, U.K., in1979.

From 1979 to 1990, he was with the Department ofMarine Sciences, Mersin, METU, Turkey, and withthe Department of Electrical Engineering, METU.In 1990, he joined the faculty of Bilkent University,

Ankara, where he is currently a Professor with the Department of Electricaland Electronics Engineering. His research interests include underwater andairborne acoustic and ultrasonic transducer design, underwater acoustics,underwater and airborne acoustic systems, acoustic microscopy, ultrasonicNDT, biomedical instrumentation, mobile communications, and spectrummanagement.

Prof. Köymen is an IET Fellow.

Abdullah Atalar (M’88–SM’90–F’07) received theB.S. degree in electrical engineering from MiddleEast Technical University (METU), Ankara, Turkey,in 1974, and the M.S. and Ph.D. degrees in electricalengineering from Stanford University, Stanford, CA,USA, in 1976 and 1978, respectively.

In 1979, he was with the Hewlett-PackardLaboratories, Palo Alto, CA, USA. From 1980 to1986, he was an Assistant Professor with METU.In 1986, he was a Chairman with the Departmentof Electrical and Electronics Engineering, Bilkent

University, Ankara, and participated in founding of the Department. In 1995,he was a Visiting Professor with Stanford University. From 1996 to 2010, hewas the provost of Bilkent University, where he is currently the Rector and aProfessor. His current research interests include micromachined devices andmicrowave electronics.

Dr. Atalar is a member of the Turkish Academy of Sciences. He was arecipient of the TUBITAK Science Award in 1994.

A. Sinan Tasdelen received the B.Sc. and M.Sc.degrees in electrical and electronics engineeringfrom Bilkent University, Ankara, Turkey, in 2004and 2007, respectively.

In 2008, he joined the Bilkent University Acousticsand Underwater Technologies Research Center,Bilkent University, where he is currently HeadResearch Engineer. His current research interestsinclude passive radar, underwater acoustics, trans-ducer design, and biomedical ultrasound.

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